Dynamical transition for a 3-component Lotka-Volterra Model with
Diffusion

The main objective of this article is to investigate the dynamical
transition for a 3-component Lotka-Volterra model with diffusion. Based
on the spectral analysis, the principle of exchange of stability
conditions for eigenvalues are obtained. In addition, when
$\delta_0<\delta_1$, the
first eigenvalues are complex, and we show that the system undergoes a
continuous or jump transition. In the small oscillation frequency limit,
the transition is always continuous and the time periodic rolls are
stable after the transition. In the case where
$\delta_0>\delta_1$, the
first eigenvalue is real. Generically, the first eigenvalue is simple
and all three types of transition are possible. In particular, the
transition is mixed if
$\int_{\Omega}e_{k_0}^3dx\neq
0$, and is continuous or jump in the case where
$\int_{\Omega}e_{k_0}^3dx= 0$.
In this case we also show that the system bifurcates to two saddle
points on $\delta<\delta_1$
as $\tilde{\theta}> 0$,
and bifurcates to two stable singular points on
$\delta>\delta_1$ as
$\tilde{\theta}< 0$ where
$\tilde{\theta}$ depends on the system
parameters.